Degree-truncated choosability of planar graphs

Assume $G$ is a graph and $k$ is a positive integer. Let $f:V(G)\to \mathbb{N}$ be defined as $f(v)=\min \{k,d_G(v)\}$. If $G$ is $f$-choosable, then we say $G$ is degree-truncated $k$-choosable. Answering a question of Richter, it was proved in [Zhou,Zhu,Zhu, Degree-truncated choice number of graphs, arXiv:2308.15853] that there exists a 3-connected non-complete planar graph that is not degree-truncated 7-choosable, and every 3-connected non-complete planar graph is degree-truncated 16-choosable. This paper improves the bounds, and proves that there exists a 3-connected non-complete planar graph that is not degree-truncated 8-choosable, and that every 3-connected non-complete planar graph is degree-truncated $12$-choosable.